A model map $Q$ for the Hopf-saddle-node (HSN) bifurcation of fixed points of diffeomorphisms is studied. The model is constructed to describe the dynamics inside an attracting invariant two-torus which occurs due to the presence of quasi-periodic Hopf bifurcations of an invariant circle, emanating from the central HSN bifurcation. Resonances of the dynamics inside the two-torus attractor yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps, where a web of hyperbolic periodic points is expected to occur inside the two-torus attractor. It is conjectured that heteroclinic intersections of the invariant manifolds of the saddle periodic points may give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. To understand this phenomenon, a simple model map of the standard two-torus is developed and studied and the relations with the starting model map $Q$ are discussed.

In this paper we will review several results of rigidity in one-dimensional dynamics and its relation to renormalization. In many one dimensional dynamical systems the existence of a topological conjugacy between two mappings implies that the restriction of the conjugacy to the attractor extends to a smooth mapping. Thus the combinatorics imposes severe restrictions on the geometry of the attractor.

In [7] we proved that a system consisting of two brusselators linearly coupled by diffusion contains strange attractors. In this paper we will give numerical examples of the chaotic behaviour predicted by the theoretical results. Some details about error analysis, not included in [7], will be supplied here.

We are interested in the asymptotic study of canard solutions in real singularly perturbed first order ODE of the form $\varepsilon u'=\Psi(x,u,a,\varepsilon)$, where $\varepsilon>0$ is a small parameter, and $a\in\mathbb R$ is a real control parameter. An operator $\Xi_\eta$ was defined to prove the existence of canard solutions. This demonstration allows us to conjecture the existence of a generalized asymptotic expansion in fractional powers of $\varepsilon$ for those solutions. In this note, we propose an algorithm that computes such an asymptotic expansions for the canard solution. Furthermore, those asymptotic expansions remain uniformly valid.

We present an example of slow-fast system which displays a full open set of initial data so that the corresponding orbit has the property that given any $\epsilon$ and $T$, it remains to a distance less than $\epsilon$ from a repulsive part of the fast dynamics and for a time larger than $T$. This example shows that the common representation of generic fast-slow systems where general orbits are pieces of slow motions near the attractive parts of the critical manifold intertwined by fast motions is false. Such a description is indeed based on the condition that the singularities of the critical set are folds. In our example, these singularities are transcritical.

In unfoldings of resonant homoclinic bellows interesting bifurcation phenomena occur: two suspensed Smale horseshoes can collide and disappear in saddle-node bifurcations (all periodic orbits disappear through saddle-node bifurcations, there are no other bifurcations of periodic orbits), or a suspended horseshoe can go through saddle-node and period-doubling bifurcations of the periodic orbits in it to create an additional ``doubled horseshoe''.

This paper concerns differential equation systems on $\mathbb R^n$ with discontinuous right--hand sides. We deal with non-smooth vector fields in $\mathbb R^n$ having a codimension-one submanifold $M$ as its discontinuity set. After a regularization of a such system and a global blow-up we are able to bring out some results that bridge the space between discontinuous systems and singularly perturbed smooth systems.

We define the center variety for families of $p:-q$ resonant polynomial vector fields and prove the correctness of the definition. We also derive an algorithm for computing the focus quantities of such vector fields.

This paper is part of the DRR-program of [4] to prove the finiteness part of Hilbert's 16th problem for quadratic vector fields by showing the finite cyclicity of 121 graphics. In this paper we prove the finite cyclicity of 4 graphics passing through a triple nilpotent point of elliptic type surrounding a center, namely the graphics $(H_7^1)$, $(F_{7a}^1)$, $(H_{11}^3)$ and $(I_{6a}^1)$. These four graphics are of pp-type, in the sense that they join two parabolic sectors of the nilpotent point. The exact cyclicity is 2 for $(H_7^1)$ and $(H_{11}^3)$. The graphics $(F_{7a}^1)$ and $(I_{6a}^1)$ occur in continuous families. Their exact cyclicity is 2 except for a discrete subset of such graphics. The method can be applied to most other graphics of the DRR-program [4] through a triple nilpotent point and surrounding a center.

New results are established here on the phase portraits and bifurcations of the kinematic model, first presented by H.K. Wilson, and by him attributed to L. Markus (unpublished). A new, self-sufficient, study which extends that of H.K. Wilson and allows an essential conclusion for the applicability of the model is reported here.

We present a multidimensional analogue of the classical Takens normal form for a nilpotent singularity of a vector field.

We consider the $K$-$\varepsilon$ model describing an expansion of a free turbulent jet. Due to the nonlinear nature of turbulent diffusion the turbulent area has a sharp boundary. We seek solutions for the energy, dissipation and momentum as power series in spatial coordinate across the jet with time-dependent coefficients. The coefficients obey a dynamical system with clearly identifiable slow and fast variables. The system is not in a standard form, which excludes rigorous methods of analysis such as centre manifold methods. We put forward a hypothesis that there exists an attracting invariant manifold for trajectories based on a few slow variables. The hypothesis is supported numerically.

We study the box dimension and Minkowski content of spiral trajectories of planar vector fields, using information about the asymptotic behaviour of iterates of the Poincaré map. An auxilliary tool is a flow-sector theorem near the weak focus, which is of a similar nature as the well known flow-box theorem. Applications include Hopf bifurcation and Liénard systems. We obtain all possible values of box dimensions of spiral trajectories around weak focus, associated with polynomial vector fields.